that there exist irrationals $$a$$ and $$b$$ such that $$a^{b}$$ is rational:
$$\sqrt{2}$$ is irrational. Let $$z=\sqrt{2}^{\sqrt{2}}$$. If $$z$$ is rational, the claim is proved. If not, then $$z^{\sqrt{2}}=2$$ proves the claim.

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